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This article describes a method of curve-fitting that involves splitting the curve into two (or more) and creating separate functions for each part. These functions are then concatenated via a linking function to create one overall function that better describes the original data than is otherwise achievable. The linking function is able to do this by separating the original two functions so that they are each active in only the relevant portion of the overall curve without the use of dummy variables. The final result is a continuous function in which it is straightforward to smooth the transition at the knot between the piecewise subfunctions. In addition, the piecewise subfunctions do not need to align at the knot since the degree of smoothing is readily controlled. All types of functions may be concatenated so that the method is flexible and relatively simple to apply. [ABSTRACT FROM AUTHOR]

AbstractDue to their flexibility and ability to incorporate non-linear relationships, Mixed-Integer Non-Linear Programming (MINLP) approaches for optimization are commonly presented as a solution tool for real-world problems. Within this context, piecewise linear (PWL) approximations of non-linear continuous functions are useful, as opposed to non-linear machine learning-based approaches, since they enable the application of Mixed-Integer Linear Programming techniques in the MINLP framework, as well as retaining important features of the approximated non-linear functions, such as convexity. In this work, we extend upon fast algorithmic approaches for modeling discrete data using PWL regression by tuning them to allow the modeling of continuous functions. We show that if the input function is convex, then the convexity of the resulting PWL function is guaranteed. An analysis of the runtime of the presented algorithm shows which function characteristics affect the efficiency of the model, and which classes of functions can be modeled very quickly. Experimental results show that the presented approach is significantly faster than five existing approaches for modeling non-linear functions from the literature, at least 11 times faster on the tested functions, and up to a maximum speedup of more than 328,000. The presented approach also solves six benchmark problems for the first time. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a new faster iterative method, called the MNiteration process, for approximating a fixed point of continuous functions on an arbitrary interval. Then, a necessary and sufficient condition for the convergence of the MN-iteration of continuous functions on an arbitrary interval is established. We also compare the rate of convergence between the proposed iteration and some other iteration processes in the literature. Specifically, our main result shows that MN-iteration converges faster than NSP-iteration to the fixed point. We finally give numerical examples to compare the result with Mann, Ishikawa, Noor, SP and NSP iterations. Our findings improve corresponding results in the contemporary literature. [ABSTRACT FROM AUTHOR]

This paper is dedicated to finding the traveling wave solutions of the loaded Burgers equation. We show how to find the solutions via the (G′/G) - expansion method which is one of the most effective ways of finding solutions. When the parameters are taken as special values, the solitary waves are also derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. This method is easy to implement using well-known software packages, which allows you to solve complex nonlinear evolution equations of mathematical physics. [ABSTRACT FROM AUTHOR]

We study the space of all continuous and increasing self-mappings of the real interval [0, 1] equipped with the topology of uniform convergence. In particular, we show that most such functions have at least two different fixed points. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of a mild solution for the fractional delay control system. Firstly, we will study the control problem. Then, we will deal with the topological structure of the solution set consisting of the compactness and  R σ  property. We will derive a mild solution to the above delay control problem by using the Laplace transform method. [ABSTRACT FROM AUTHOR]

We present a streamlined, slightly modified version, in the two-variable situation, of a beautiful, but not so well known, theory by Bogel [1, 2], already from the 1930s, on an alternative higher dimensional calculus of real functions, a double calculus, which includes many two-variable extensions of classical results from single variable calculus, such as Rolle's theorem, Lagrange's mean value theorem, Cauchy's mean value theorem, Fermat's extremum theorem, the first derivative test, and the first and second fundamental theorems of calculus. [ABSTRACT FROM AUTHOR]

Some sufficient conditions are presented for the continuity of the mapping f : ⟨ X , τ X * ⟩ → ⟨ Y , τ Y * ⟩ , where τ X * and τ Y * are the topologies induced by the local function on 푋 and 푌, resp. under the assumption that the mapping from ⟨ X , τ X ⟩ to ⟨ Y , τ Y ⟩ is continuous. Further, we consider open and closed functions in the cases in which the open (or closed) mapping is preserved through the domain and codomain "idealization". Through several examples, we illustrate that the considered conditions cannot be weakened. [ABSTRACT FROM AUTHOR]

The focus of this paper is to study unbounded variation functions from the perspective of Hölder conditions. Three special unbounded variation functions have been constructed. The first is a continuous function of unbounded variation that satisfies the Hölder condition of a given order and the second is a continuous function of unbounded variation that does not satisfy the Hölder condition of any order. The third is a continuous function of unbounded variation defined on any sub-interval of the interval I. Then, specific fractal dimension analysis of the above functions and relevant conclusions have been investigated. Finally, combining functional analysis and reinforcement learning, the convergence of reinforcement learning algorithms can be proved in unified framework. [ABSTRACT FROM AUTHOR]

A representation formula for locally defined operators acting between Banach spaces of continuous functions of bounded variation in the Waterman sense is presented. Moreover, the Nemytskij composition operators will be investigated and some consequences for locally bounded as well as uniformly bounded local operators will be given. [ABSTRACT FROM AUTHOR]

We unify various étale groupoid reconstruction theorems such as the following: • Kumjian and Renault's reconstruction from a groupoid C*-algebra; • Exel's reconstruction from an ample inverse semigroup; • Steinberg's reconstruction from a groupoid ring; • Choi, Gardella and Thiel's reconstruction from a groupoid L p {L^{p}} -algebra. We do this by working with certain bumpy semigroups S of functions defined on an étale groupoid G. The semigroup structure of S together with the diagonal subsemigroup D then yields a natural domination relation ≺ {\prec} on S. The groupoid of ≺ {\prec} -ultrafilters is then isomorphic to the original groupoid G. [ABSTRACT FROM AUTHOR]

In this paper, we mainly construct two continuous functions defined on I = [ 0 , 1 ] which have only one unbounded variation point, and show that both Box and Hausdorff dimensions of these functions are 1. We prove that their Riemann–Liouville fractional integrals are bounded variation function. We also show that these functions repair inaccuracies of the example in [N. Liu and K. Yao, Fractal dimension of a special continuous function, Fractals26(4) (2018) 1850048; Y. S. Liang, Definition and classification of one-dimensional continuous functions with unbounded variation, Fractals25(5) (2017) 1750048]. In the proofs, we mainly use Leibniz's alternating series test and the properties of logarithmic function. [ABSTRACT FROM AUTHOR]

A three-hidden-layer neural network with super approximation power is introduced. This network is built with the floor function (⌊ x ⌋), the exponential function (2 x), the step function (1 x ≥ 0), or their compositions as the activation function in each neuron and hence we call such networks as Floor-Exponential-Step (FLES) networks. For any width hyper-parameter N ∈ N + , it is shown that FLES networks with width max { d , N } and three hidden layers can uniformly approximate a Hölder continuous function f on [ 0 , 1 ] d with an exponential approximation rate 3 λ (2 d) α 2 − α N , where α ∈ (0 , 1 ] and λ > 0 are the Hölder order and constant, respectively. More generally for an arbitrary continuous function f on [ 0 , 1 ] d with a modulus of continuity ω f (⋅) , the constructive approximation rate is 2 ω f (2 d) 2 − N + ω f (2 d 2 − N). Moreover, we extend such a result to general bounded continuous functions on a bounded set E ⊆ R d . As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of ω f (r) as r → 0 is moderate (e.g., ω f (r) ≲ r α for Hölder continuous functions), since the major term to be concerned in our approximation rate is essentially d times a function of N independent of d within the modulus of continuity. Finally, we extend our analysis to derive similar approximation results in the L p -norm for p ∈ [ 1 , ∞) via replacing Floor-Exponential-Step activation functions by continuous activation functions. [ABSTRACT FROM AUTHOR]

The present paper introduces and studies the concepts of K-outer approximation and K-inner approximation for a monotone function μ defined on a D-poset P, by a subfamily K of P. Some desirable properties of K-approximable functions are established and it is shown that the family of all elements of P that possess K-approximation, forms a lattice and is closed under orthosupplementation. We have proved that a submodular measure on a suitable subfamily of P having K-outer approximation can be extended to a function that has K-outer approximation, and a tight function that has K-inner approximation can be extended to a function having K-inner approximation. [ABSTRACT FROM AUTHOR]

The problem of characterizing continuous functions belonging to each A-ring containing a fixed Darboux function is fully solved for a compact interval domain and solved with some minor exception for a real line domain. Many properties of generated A-rings and their continuous functions are stated. [ABSTRACT FROM AUTHOR]

In this research the aims we introduction of new concepts in topological spaces named DG- topological spaces topology associated with digraphs induced by new open set called DG-open set and investigated some properties of this concepts by the closed connection between topology and digraphs. [ABSTRACT FROM AUTHOR]

Rock mass classification has been playing an important role in the analyses of civil engineering in recent years. The quality evaluation of slope rock mass becomes essential for slope stability assessment. Neither continuous expression and comparable function are rarely taken into consideration in rock mass classification, nor is the shortage in the previous quality evaluation of slop rock mass. Through constructing the benchmark intervals of evaluation indicators; mathematical models between quantized value and relative quantized value which is a new proposed parameter; and linear weighted function, the interval continuous mathematical models (ICMM) for quality evaluation of slope rock mass are presented. The application result shows that ICMM is effective for improving the comparability of rock mass quality of slopes by unifying evaluation indicators. On the other hand, the ICMM realizes the continuous expression of rock mass quality of slopes by removing the step-like features caused by the pre-set rating criteria of each evaluation indicator and rock mass quality classification, guiding the follow-up construction and protection of slopes, and ensuring the balance between economy and safety. Therefore, the ICMM is more scientific and reasonable than extant classification systems for evaluating rock mass quality of slopes. [ABSTRACT FROM AUTHOR]

: In this paper, author introduces some new integral inequalities by using the Caputo-Fabrizio (CF) fractional integral and functions with the same sense of variation. Also an open problem concerning an integral inequality is discussed. [ABSTRACT FROM AUTHOR]

A function f : X → ℝ defined on a topological space X is called returning if for any point x ∈ X there exists a positive real number Mx such that for every path-connected subset Cx ⊂ X containing the point x and any y ∈ Cx ∖ {x} there exists a point z ∈ Cx ∖ {x, y} such that |f(z)| ≤ max{Mx, |f(y)|}. A topological space X is called path-inductive if a subset U ⊂ X is open if and only if for any path γ : [0, 1] → X the preimage γ–1(U) is open in [0, 1]. The class of path-inductive spaces includes all first-countable locally path-connected spaces and all sequential locally contractible spaces. We prove that a function f : X → ℝ defined on a path-inductive space X is continuous if and only if it is returning and has closed graph. This implies that a (weakly) Świątkowski function f : ℝ → ℝ is continuous if and only if it has closed graph, which answers a problem of Maliszewski, inscribed to Lviv Scottish Book. [ABSTRACT FROM AUTHOR]

We examine images of Bernstein sets via continuous mappings. Among other results, we prove that there exists a continuous function f : ℝ → ℝ {f\colon\mathbb{R}\to\mathbb{R}} that maps every Bernstein subset of ℝ {\mathbb{R}} onto the whole real line. This gives the positive answer to a question of Osipov. [ABSTRACT FROM AUTHOR]

In this paper, we first give a necessary and sufficient condition for convergence of Picard-S iteration process to a fixed point of continuous functions on an arbitrary interval and prove equivalence of Picard-S and P iterative processes. We also compare the rate of convergence between Picard-S and some others iteration processes in the literature. Finally, some numerical examples for comparing the rate of convergence of those methods are also given. [ABSTRACT FROM AUTHOR]

The aim of this expository article is to present recent developments in the centuries-old discussion on the interrelations between continuous and differentiable real valued functions of one real variable. The truly new results include, among others, the Dn-Cn interpolation theorem: For every n-times differentiable f : ℝ → ℝ and perfect P ⊂ ℝ, there is a Cn function g : ℝ → ℝ such that f ↾ P and g ↾ P agree on an uncountable set and an example of a differentiable function F : ℝ → ℝ (which can be nowhere monotone) and of compact perfect X ⊂ ℝ such that F'(x) = 0 for all x ∈ X while F[X] = X. Thus, the map f = F ↾ X is shrinking at every point though, paradoxically, not globally. However, the novelty is even more prominent in the newly discovered simplified presentations of several older results, including a new short and elementary construction of everywhere differentiable nowhere monotone h: ℝ → ℝ and the proofs (not involving Lebesgue measure/integration theory) of the theorems of Jarník--Every differentiable map f : P → ℝ, with P ⊂ ℝ perfect, admits differentiable extension F : ℝ → ℝ--and of Laczkovich--For every continuous g : ℝ → ℝ there exists a perfect P ⊂ ℝ such that g ↾ P is differentiable. The main part of this exposition, concerning continuity and first-order differentiation, is presented in a narrative that answers two classical questions: To what extent must a continuous function be differentiable? and How strong is the assumption of differentiability of a continuous function? In addition, we give an overview of the results concerning higher-order differentiation. This includes the Whitney extension theorem and the higher-order interpolation theorems related to the Ulam-Zahorski problem. Finally, we discuss the results concerning smooth functions that are independent of the standard axioms of ZFC set theory. We close with a list of currently open problems related to this subject. [ABSTRACT FROM AUTHOR]

For a Tychonoff space X, we will denote by USCp(X) (B1(X)) the set of all real-valued upper semicontinuous functions (the set of all Baire functions of class 1) defined on X endowed with the pointwise convergence topology. In this paper we describe a class of Tychonoff spaces X for which the space USCp(X) is sequentially separable. Unexpectedly, it turns out that this class coincides with the class of spaces for which a stronger form of the sequential separability for the space B1(X) holds. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the computation of the antiderivatives on R of a special class of continuous periodic functions. Finally, some applications of the main result are presented. [ABSTRACT FROM AUTHOR]

Let X be a bounded subset of Rd and write Cu(X) for the set of uniformly continuous functions on X equipped with the uniform norm. The lower and upper box dimensions, denoted by dim̲B(graph(f)) and dim¯B(graph(f)), of the graph graph(f) of a function f∈Cu(X) are defined by dim̲B(graph(f))=limδ↘0inflogNδ(graph(f))-logδ,dim¯B(graph(f))=limδ↘0suplogNδ(graph(f))-logδ,where Nδ(graph(f)) denotes the number of δ-mesh cubes that intersect graph(f).Hyde et al. have recently proved that the box counting function (∗)logNδ(graph(f))-logδof the graph of a typical function f∈Cu(X) diverges in the worst possible way as δ↘0. More precisely, Hyde et al. showed that for a typical function f∈Cu(X), the lower box dimension of the graph of f is as small as possible and if X has only finitely many isolated points, then the upper box dimension of the graph of f is as big as possible.In this paper we will prove that the box counting function (*) of the graph of a typical function f∈Cu(X) is spectacularly more irregular than suggested by the result due to Hyde et al. Namely, we show the following surprising result: not only is the box counting function in (*) divergent as δ↘0, but it is so irregular that it remains spectacularly divergent as δ↘0 even after being “averaged" or “smoothened out" using exceptionally powerful averaging methods including all higher order Hölder and Cesàro averages and all higher order Riesz-Hardy logarithmic averages. For example, if the box dimension of X exists, then we show that for a typical function f∈Cu(X), all the higher order lower Hölder and Cesàro averages of the box counting function (*) are as small as possible, namely, equal to the box dimension of X, and if, in addition, X has only finitely many isolated points, then all the higher order upper Hölder and Cesàro averages of the box counting function (*) are as big as possible, namely, equal to the box dimension of X plus 1. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and extend the concept of the weighted transform and its inverse to functions of two variables. We show that they can be applied to image compression and decompression. The quality of the reconstructed image by inverse weighted transform is also compared with some known methods such as the fuzzy transform method and the standard JPEG method. [ABSTRACT FROM AUTHOR]

Three types of N-open sets are defined and investigated in bitopological spaces, and via them several compactness are introduced. Several relationships, examples and counter-examples regarding the new concepts are given. [ABSTRACT FROM AUTHOR]

We present some results involving the Hyers-Ulam stability of some intermediary point arising from the mean value theorems. First, we make some considerations about the stability of Lagrange mean point and we prove a stability result involving Cauchy mean point. Also, we establish some conditions to the stability of the intermediary point defined by some recent theorems due to Wachnichi or Tong. We investigate similar problems to the integral version of these theorems. We complete with Hyers-Ulam stability of the intermediary point arising from second integral mean value theorem. Also, we present a new integral mean theorem and its stability. [ABSTRACT FROM AUTHOR]

In this paper, we consider S-iteration to find fixed points of continuous mappings on an arbitrary interval. We give some necessary and sufficient conditions for the convergence of this iteration. Also, we proved that the rate of convergence of S-iteration is better than some other iterations for continuous and nondecreasing mappings. It is also noted that the method of proof of Lemma 3 using S-iteration is slightly different from that using the iteration schemes like Mann. [ABSTRACT FROM AUTHOR]

Let ? be a unimodular complex number, and let k be an integer. Then ?Ak is an isometry for any isometry A of a complex Banach space. It is shown that if f is an analytic function on the unit circle sending an isometry to an isometry for any norm, then f has the form z ? ?zk for some unimodular ? and integer k. The same conclusion on f can be deduced if f is merely continuous and preserves the isometries of some special classes of norms on a fixed finite-dimensional complex Banach space. The result is extended to real Banach spaces X with dimX = 4, and it is shown that one cannot get the same conclusion on f if dimX <4. Further extensions of these results are also considered. [ABSTRACT FROM AUTHOR]

Let γ be a unimodular complex number, and let k be an integer. Then γAk is an isometry for any isometry A of a complex Banach space. It is shown that if f is an analytic function on the unit circle sending an isometry to an isometry for any norm, then f has the form z 7→ γzk for some unimodular γ and integer k. The same conclusion on f can be deduced if f is merely continuous and preserves the isometries of some special classes of norms on a fixed finite-dimensional complex Banach space. The result is extended to real Banach spaces X with dim X ≥ 4, and it is shown that one cannot get the same conclusion on f if dim X < 4. Further extensions of these results are also considered. [ABSTRACT FROM AUTHOR]

Kaplansky [7] proved that C K ( X ) is the intersection of all free maximal ideals in C ( X ) in the case of discrete X , and asked whether the equality holds in general. In this paper we prove that C K ( X ) coincides with the intersection of all free maximal ideals if and only if every open hemicompact z -compact (i.e., every zero-set contained in it is compact) subset of X is relatively compact or equivalently, every open Lindelöf z -compact subset of X is relatively compact. We conclude that the equality holds whenever X is a strongly isocompact space. [ABSTRACT FROM AUTHOR]

We study cardinal invariants related to quotients in the case of the complement in ℝR of families of continuous, quasi-continuous, cliquish and Darboux functions. [ABSTRACT FROM AUTHOR]

In this work, we introduce a transformation via the concept of a weighted partition. We have shown that real-valued continuous functions defined on unbounded interval $$[0,\infty )$$ which vanish at infinity, can be approximated by this transformation with an arbitrary precision. We also give an approximation for integrable functions on $$[0,\infty )$$ . Some examples, which support our main results, are given. [ABSTRACT FROM AUTHOR]

In this paper we study a sufficient conditions for continuous and θ α -continuous extensions of f to space X for an image space Y with different separation axioms. [ABSTRACT FROM AUTHOR]

Some classes of metric spaces satisfying properties stronger than completeness but weaker than compactness have been studied by many authors over the years. One such significant family consists of those metric spaces on which every real-valued continuous function is uniformly continuous, which are widely known as Atsuji spaces or UC spaces. Recently in 2014, two new kinds of complete metric spaces are introduced, namely Bourbaki-complete and cofinally Bourbaki-complete metric spaces, whose idea has come from some new classes of sequences acting as generalizations of Cauchy sequences. Our major goal is to give several new equivalent conditions for metric spaces whose completions are one of the aforesaid spaces, especially in terms of some functions, sequences and geometric functionals. [ABSTRACT FROM AUTHOR]

A fixed point compactification of a locally compact noncompact group G is a faithful semigroup compactification S such that $$ap=pa=p$$ for all $$p\in S\setminus G$$ and $$a\in G$$ . Since the right translations are continuous, the remainder of a fixed point compactification is a right zero semigroup. Among all fixed point compactifications of G there is a largest one, denoted $$\theta G$$ . We show that if G is $$\sigma $$ -compact, then $$\theta G\setminus G$$ contains a copy of $$\beta \omega \setminus \omega $$ . In contrast, if G is not $$\sigma $$ -compact, then $$\theta G$$ is the one-point compactification. [ABSTRACT FROM AUTHOR]

In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are . The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be when is self-similar. [ABSTRACT FROM AUTHOR]

In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15], [12], [10], and [11]. [ABSTRACT FROM AUTHOR]

The research on the collision between a spinning ball and a table, as a general issue for the trajectory analysis of the spinning-flying objects, is important but complicated. Traditionally, it is studied either by simplifying the collision process as a black box in which the ball’s motion state changes abruptly before and after collision, or by assuming that the coefficient of restitution and the coefficient of friction are simply piecewise linear or even constant during the whole collision process. In this paper, we first analyze the mechanism of the collision dynamics and conclude from mathematical derivations that the collision between a ping-pong ball and a table is a continuous transition process and its collision duration is a constant irrelevant to the ball’s motion state. These two conclusions are also verified by the visual measurement using an ultrahigh-speed camera. Then, we propose a novel rebound model using the mean value theorem, momentum theorem, and angular momentum theorem. The novelty of the proposed model is that the forces and the parameters in it are formulated as the continuous functions related to the ball’s motion state, which is more accurate, rather than as constants. The integrations of these functions over time can be formulated by using the mean value theorem, and their expressions are learned using multilayer perception with a large data set collected by the position vision system and the pan-tilt vision system. The experimental results verify the effectiveness and accuracy of the proposed model. [ABSTRACT FROM AUTHOR]

In this note we shall prove that for a continuous function φ : Δ → Rn, where Δ ⊂ R, the paratingent of φ at a ∈ Δ is a non-empty and compact set in Rn if and only if φ satisfies Lipschitz condition in a neighbourhood of a. Moreover, in this case the paratingent is a connected set. [ABSTRACT FROM AUTHOR]

We establish a necessary and sufficient condition on a continuous function on [-1,1] under which the family of functions on the unit sphere 핊 d-1 constructed in the described manner is fundamental in C(핊 d-1). In our construction of functions and proof of the result, we essentially use Dunkl harmonic analysis. [ABSTRACT FROM AUTHOR]

In this note we shall prove that for a continuous function φ: Δ → Rn, where Δ ⊂ R, the paratingent of φ at a ∊ Δ is a non-empty and compact set in Rn if and only if φ satisfies Lipschitz condition in a neighbourhood of a. Moreover, in this case the paratingent is a connected set. [ABSTRACT FROM AUTHOR]

Denote by [0,ω1) the set of countable ordinals, equipped with the order topology, let L0 be the disjoint union of the compact ordinal intervals [0, α] for α countable, and consider the Banach spaces C0[0,ω1) and C0(L0) consisting of all scalar-valued, continuous functions which are defined on the locally compact Hausdorff spaces [0,ω1) and L0, respectively, and which vanish eventually. Our main result states that a bounded, linear operator T between any pair of these two Banach spaces fixes an isomorphic copy of C0(L0) if and only if the identity operator on C0(L0) factors through T, if and only if the Szlenk index of T is uncountable. This implies that the set ...C0(L0)(C0(L0)) of C0(L0)-strictly singular operators on C0(L0) is the unique maximal ideal of the Banach algebra B(C0(L0)) of all bounded, linear operators on C0(L0), and that SC0(L0) (C0[0,ω1) is the second-largest proper ideal of B(C0[0,ω1)) . Moreover, it follows that the Banach space C0(L0) is primary and complementably homogeneous. [ABSTRACT FROM AUTHOR]

Let $$E$$ be a bounded subset of real line which contains its infimum and supremum. In this paper, we have defined the $$\phi -$$ transform and its inverse, where $$\phi $$ is a function from $$E$$ into $$(0,1]$$ . We will have shown that real-valued integrable functions on $$[a, b]$$ and real-valued continuous functions on $$E$$ can be approximated by this transformation with an arbitrary precision. [ABSTRACT FROM AUTHOR]